QUESTION 5: Given: $$ -58 ;-38 ;-21 ;-7:- $$ 5.1 Writo down the values of the next two terms 5.2 Determine $$ T n $$. 5.3 Caiculate the largest term in the pattorn above. 5.4 Provide a reason why the sequence will not have a minimum value. 5.5 Which term will be the first to be less than - 400?

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Let's analyze the given sequence and address each part of the question systematically. **Given Sequence:** $$ -58, -38, -21, -7, \ldots $$ ### 5.1 **Write Down the Values of the Next Two Terms** **Analysis:** - **First differences:** $$ -38 - (-58) = 20,\quad -21 - (-38) = 17,\quad -7 - (-21) = 14 $$ - The differences between terms decrease by 3 each time: $$ 20,\ 17,\ 14,\ 11,\ 8,\ \ldots $$ - **Next differences:** $$ 11,\ 8 $$ - **Next terms:** $$ -7 + 11 = 4,\quad 4 + 8 = 12 $$ **Answer:** The next two terms are **4** and **12**. --- ### 5.2 **Determine $$ T_n $$ (The $$ n^{th} $$ Term)** **Approach:** The sequence is generated by adding differences that decrease by 3 each time. This suggests a quadratic relationship. Let $$ T_n = an^2 + bn + c $$. **Using Initial Terms:** $$ \begin{cases} T_1 = -58: & a + b + c = -58 \ T_2 = -38: & 4a + 2b + c = -38 \ T_3 = -21: & 9a + 3b + c = -21 \ \end{cases} $$ **Solving the Equations:** 1. Subtract the first equation from the second: $$ 3a + b = 20 $$ 2. Subtract the second equation from the third: $$ 5a + b = 17 $$ 3. Subtract these two results: $$ 2a = -3 \Rightarrow a = -1.5 $$ 4. Substitute $$ a $$ into $$ 3a + b = 20 $$: $$ -4.5 + b = 20 \Rightarrow b = 24.5 $$ 5. Substitute $$ a $$ and $$ b $$ into $$ a + b + c = -58 $$: $$ -1.5 + 24.5 + c = -58 \Rightarrow c = -81 $$ **Final Formula:** $$ T_n = -1.5n^2 + 24.5n - 81 $$ **Answer:** The $$ n^{th} $$ term is $$ T_n = -1.5\,n^2 + 24.5\,n - 81 $$. --- ### 5.3 **Calculate the Largest Term in the Pattern** **Approach:** Since $$ T_n = -1.5n^2 + 24.5n - 81 $$ is a downward-opening parabola, its maximum occurs at the vertex. **Vertex Formula:** $$ n = -\frac{b}{2a} = -\frac{24.5}{2(-1.5)} = \frac{24.5}{3} \approx 8.166 $$ Thus, the maximum occurs around $$ n = 8 $$. **Calculating $$ T_8 $$:** $$ T_8 = -1.5(8)^2 + 24.5(8) - 81 = -96 + 196 - 81 = 19 $$ **Answer:** The largest term in the sequence is **19** (which is the 8th term). --- ### 5.4 **Why the Sequence Will Not Have a Minimum Value** **Explanation:** The general term $$ T_n = -1.5n^2 + 24.5n - 81 $$ is a quadratic function that opens downward (since the coefficient of $$ n^2 $$ is negative). As $$ n $$ increases, the term $$ -1.5n^2 $$ dominates, causing $$ T_n $$ to decrease without bound. **Answer:** Because as $$ n $$ increases, the terms become indefinitely negative, the sequence decreases without limit and thus has no minimum value. --- ### 5.5 **Which Term Will Be the First to Be Less Than -400?** **Objective:** Find the smallest integer $$ n $$ such that $$ T_n < -400 $$. **Set Up the Inequality:** $$ -1.5n^2 + 24.5n - 81 < -400 $$ $$ -1.5n^2 + 24.5n + 319 < 0 $$ Multiply by -1 (and reverse the inequality sign): $$ 1.5n^2 - 24.5n - 319 > 0 $$ Multiply by 2 to eliminate decimals: $$ 3n^2 - 49n - 638 > 0 $$ **Solving the Quadratic Equation:** $$ 3n^2 - 49n - 638 = 0 $$ $$ n = \frac{49 \pm \sqrt{49^2 + 4 \times 3 \times 638}}{2 \times 3} = \frac{49 \pm \sqrt{10057}}{6} \approx \frac{49 \pm 100.28}{6} $$ Only the positive root is relevant: $$ n \approx \frac{149.28}{6} \approx 24.88 $$ Thus, the first integer $$ n $$ where $$ T_n < -400 $$ is **25**. **Verification:** $$ T_{24} = -1.5(24)^2 + 24.5(24) - 81 = -864 + 588 - 81 = -357 \quad (\text{Not less than -400}) $$ $$ T_{25} = -1.5(25)^2 + 24.5(25) - 81 = -937.5 + 612.5 - 81 = -406 \quad (\text{Less than -400}) $$ **Answer:** The **25th term** is the first term less than –400. **5.1 Next Two Terms:** 4 and 12 **5.2 $$ T_n $$:** $$ T_n = -1.5n^2 + 24.5n - 81 $$ **5.3 Largest Term:** 19 (8th term) **5.4 No Minimum Value:** The sequence decreases without bound as $$ n $$ increases. **5.5 First Term Less Than -400:** 25th term

QUESTION 5:
Given:
5.1 Writo down the values of the next two terms
5.2 Determine .
5.3 Caiculate the largest term in the pattorn above.
5.4 Provide a reason why the sequence will not have a minimum value.
5.5 Which term will be the first to be less than - 400?

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Solution

5.1 Write Down the Values of the Next Two Terms

5.2 Determine (The Term)

5.3 Calculate the Largest Term in the Pattern

5.4 Why the Sequence Will Not Have a Minimum Value

5.5 Which Term Will Be the First to Be Less Than -400?

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QUESTION 5: Given: 5.1 Writo down the values of the next two terms 5.2 Determine . 5.3 Caiculate the largest term in the pattorn above. 5.4 Provide a reason why the sequence will not have a minimum value. 5.5 Which term will be the first to be less than - 400?